BGE: How to make an object leave [skid]marks on another object
The best way to do this (and I tested) would be to have your trigger object (tires) rigged with an Edit object actuator set to add object : Activate upon a collision with ground or a sharp turn (Make sure True level triggering is on).

Set the time to 30, and the target object to be a plane textured like a portion or you tires tread. Alpha channel turn down.

I got good effects with a rolling monkey head, so hey; why not a car?